Linear independence

In the theory of vector spaces the concept of linear dependence and linear independence of the vectors in a subset of the vector space is central to the definition of dimension. A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the other vectors. If no vector in the set can be written in this way, then the vectors are said to be linearly independent.[1]

A vector space can be of finite dimension or infinite dimension depending on the number of linearly independent basis vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space are linearly dependent are central to determining a set of basis vectors for a vector space.

The vectors in a subset S=(v1,v2,...,vk) of a vector spaceV are said to be linearly dependent, if there exist a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that

where zero denotes the zero vector.

Notice that if not all of the scalars are zero, then at least one is non-zero, say a1, in which case this equation can be written in the form

Thus, v1 is shown to be a linear combination of the remaining vectors. It is worth noting that non-zero a1 and the equation defining linear dependence together imply that at least one other scalar ai is non-zero.

The vectors in a set T=(v1,v2,...,vn) are said to be linearly independent if the equation

can only be satisfied by ai=0 for i=1,..., n. This implies that no vector in the set can be represented as a linear combination of the remaining vectors in the set. In other words, a set of vectors is linearly independent if the only representations of 0 as a linear combination of its vectors is the trivial representation in which all the scalars ai are zero.[2]

In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linearly dependence as follows. More generally, let V be a vector space over a fieldK, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a family {aj | j∈J} of elements of K, not all zero, such that

where the index set J is a nonempty, finite subset of I.

A set X of elements of V is linearly independent if the corresponding family {x}x∈X is linearly independent. Equivalently, a family is dependent if a member is in the linear span of the rest of the family, i.e., a member is a linear combination of the rest of the family. The trivial case of the empty family must be regarded as linearly independent for theorems to apply.

A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. For example, the vector space of all polynomials in x over the reals has the (infinite) subset {1, x, x2, ...} as a basis.

A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 5 miles northeast of here." Although this last statement is true, it is not necessary.

In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary.

Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, n linearly independent vectors are required to describe any location in n-dimensional space.

An alternative method uses the fact that n vectors in are linearly independentif and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.

In this case, the matrix formed by the vectors is

We may write a linear combination of the columns as

We are interested in whether AΛ = 0 for some nonzero vector Λ. This depends on the determinant of A, which is

Since the determinant is non-zero, the vectors (1, 1) and (−3, 2) are linearly independent.

Otherwise, suppose we have m vectors of n coordinates, with m < n. Then A is an n×m matrix and Λ is a column vector with m entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of n equations. Consider the first m rows of A, the first m equations; any solution of the full list of equations must also be true of the reduced list. In fact, if 〈i1,...,im〉 is any list of m rows, then the equation must be true for those rows.

Furthermore, the reverse is true. That is, we can test whether the m vectors are linearly dependent by testing whether

for all possible lists of m rows. (In case m = n, this requires only one determinant, as above. If m > n, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.

A linear dependence among vectors v1, ..., vn is a tuple (a1, ..., an) with nscalar components, not all zero, such that

If such a linear dependence exists, then the n vectors are linearly dependent. It makes sense to identify two linear dependences if one arises as a non-zero multiple of the other, because in this case the two describe the same linear relationship among the vectors. Under this identification, the set of all linear dependences among v1, ...., vn is a projective space.

The covariance is sometimes called a measure of "linear dependence" between two random variables. That does not mean the same thing as in the context of linear algebra. When the covariance is normalized, one obtains the correlation matrix. From it, one can obtain the Pearson coefficient, which gives us the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.